Your result is correct,  and it can be easily explained:

the number a(n) must cover all the qualifying subsets of n-1 integers i.e,  a(n-1) plus all the qualifying subsets of n-2 elements, where to each of them integer n was added (the integer n-1 is absent!).

a(n)= a(n-1) + a(n-2)

a(1)=2=f(3)   so  a(n)=f(n+2)